Stockholm School of Economics in Riga Financial Economics, Spring 2010 Jevgenijs Babaicevs
Problem Set VI: The Black-Merton-Scholes Option Pricing Model Problem 1: Introduction to the BMS Model a) Peteris is interested in purchasing a European call on a new hot high-tech stock Moodle Inc. The call has a strike price of $100 and expires in 90 days. Currently, the stock is trading at a price of $120 and has a standard deviation of 40% per year. The continuously compounded riskfree interest rate is 6.18% per year. (i) Use the Black-Merton-Scholes formula to calculate the price of the call. (ii) Compute the price of the put with the same strike and expiration date.
b) Assume that you own a small IT company. You have just received an offer to buy your company from a large, publicly traded firm, Cisco Systems (CS). Under the terms of the offer, you will receive 1 million shares of CS. CS stock currently trades at LVL 25 per share. You may decide to sell the shares of CS that you will receive in the market at any time. However, as part of the offer, CS also agrees that at any time during the next year, it will buy the shares back from you for LVL 25 per share if you want. Assume the current continuously compounded riskfree interest rate is 6.18% a year, and the volatility of CS stock is 30% per annum. Assume further that no dividends are paid. (i) Calculate the value of the offer and explain why it is worth more than LVL 25 million?
Problem 2: Option Pricing in a Discrete and Continuous Time The current stock price of Paalzow Oil (PO) is SEK 200. The standard deviation is 22.3% per year, and the continuously compounded interest rate is 21% a year. A one-year call option on PO has an exercise price of SEK 180. a) Use the Black-Merton-Scholes model to value the call option on PO. b) Calculate the up-step and down-step that you would use if you valued the PO option with the one-period Binomial Option Pricing Model. Then value the option using this model! c) Re-value the option by using the two-period Binomial Option Pricing Model. d) Use your answer to part c) to calculate the option delta (i) today, (ii) next period if the stock price rises, and (iii) next period if the stock price falls. Show at each point how you would replicate a call option with a levered investment in the company’s stock.
1
Problem 3: Collateralized Debt Obligations Note: This is a slightly modified version of Problem 5 from Re-Exam 2008. Diana P. has just informed you that your summer internship is going to be at SweetBank this year. The bank is planning to offer three types of claims, which mature in one year, on the value of its underlying portfolio consisting of commercial loans issued to residents of the Independent Republic of Junkbondia. The three claims differ in the seniority of their claims to the value of the underlying portfolio, with tranche 1 T1 being most senior and tranche 3 T3 being most junior. The claims of the tranches are paid in the order of their seniority: T1 pays the value of the underlying loan portfolio up to a maximum of D1 , and is paid first; T2 pays the excess value of the loan portfolio up to a maximum of D2 , after T1 has been paid; T3 is a claim to the residual value of the loan portfolio after the claims of the first two tranches are satisfied. a) Using real options language, draw separate position diagrams for the three tranches as a function of the value of the underlying loan portfolio V . Do not forget to denote the critical values on the axes! b) Explain the character (type) of the options associated with each of the claims! Express the value of each of the tranches as a function of V and C E - the price of a European call option written on the value of the underlying loan portfolio with an exercise price of E . c) Your summer internship has slowly turned to your full employment and you are now a CFO of SweetBank. As part of your compensation package you receive several units of one of the tranches. After taking over as CFO your first task is to decide how many loans N to include in the underlying portfolio in such a way that the market value of your compensation package is maximized! You know that each of the loans is equally weighted in the portfolio. The return on each of the loans has an annual variance of 2 and any two returns have a correlation coefficient of 0 1 . (i) You are given several units of tranche T1 . How many loans will you decide to include in the underlying portfolio? (ii) You are given several units of tranche T3 . How many loans will you decide to include in the underlying portfolio? In both cases, show your reasoning explicitly! d) Suppose the board of SweetBank has decided to compensate you with units of tranche T3 and you have optimally chosen the number of loans to include in the portfolio. Moreover, the maturity of the structured products has been extended from one year to two years. Enthused by this change, you do some research and discover that at the annual frequency the returns on some loans exhibit momentum (Loans-AA), while others exhibit mean-reversion (Loans-BB). The annual variance of returns of both types of loans is still 2 . (i) Which type of loans will you choose to include in the underlying portfolio?
2
Additional Problem for Your own Study Problem 4: BMS Model and Land-Owners Alexandre owns a one-year call option on 1 acre of real estate in Paris. The exercise price is $2 million and the current, appraised market value of the land is $1.7 million. The land is currently used as a parking lot, generating just enough money to cover real estate taxes. The annual standard deviation is 15% and the continuously compounded interest rate is 12%. a) Use the Black-Merton-Scholes formula to calculate the price of a call belonging to Alexandre? b) Assume now that the land is occupied by a warehouse generating rents of $150,000 after real estate taxes and all other expenses. The value of the land plus warehouse is again $1.7 million. Alexandre has a European call option. What is the value of the call now? Answer:
b) C ' USD 28, 640.445
a) C USD 71,167.285
3
